I will discuss some recent forays into some counting problems for free objects. I will focus on free inverse semigroups and free regular $*$-semigroups. I will first discuss recent results joint with M. Kambites, N. Szakács, and R. Webb giving a precise rate of exponential growth of the free inverse monoid of arbitrary (finite) rank, which turns out to be given by a surprisingly complicated but algebraic number.

Given a group $G$ acting on a set $X$, an element $g$ of $G$ is called a derangement if it acts without fixed points on $X$. The Boston-Shalev conjecture, proved by Fulman and Guralnick, asserts that in a finite simple group $G$ acting transitively on $X$, the proportion of derangements is at least some absolute constant $c>0$. After giving an introduction to the subject, I will present a version of the conjecture for the proportion of   *conjugacy classes* containing derangements in finite groups of Lie type.

The length of a finite system of generators for a finite-dimensional algebra over a field is the least positive integer $k$ such that the products of length not exceeding $k$ span this algebra as a vector space.The maximum length for the systems of generators of an algebra is called the length of this algebra. Length function is an important invariant widely used to study finite dimensional algebras since 1959. The length evaluation is a difficult problem.  For example, the length of the full matrix algebra is still  unknown.

I will present some results that were obtained in collaboration with Csaba Schneider (Universidade Federal de Minas Gerais) concerning groups that have a solvable subgroup of prime-power index. Under weak conditions such groups are solvable and, when they are not, the index of their solvable radical is asymptotically small. 
 

Keywords: Solvability and Fermat primes

Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma $-representations of $G$ into algebras over a field $\kappa .$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^{\sigma}G$   with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma}G.$   Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral se

The simple positive energy representations for an affine vertex algebras associated to higher rank Lie algebras of type $A$ and admissible levels whose top spaces are induced modules of cuspidal representation of type $A$ are extensively studied. Note that all simple weight modules with finite-dimensional weight spaces are classified. However, the general classification of simple admissible weight modules of type $A$ is not know. In this talk, we explain the recent developments in the general open problem.